Find a Plane Containing Two Given Lines

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Recall that two intersecting lines determine a plane. We often have in geology the problem of finding a plane that contains two intersecting lines or is parallel to two non-intersecting lines. The method we use is similar to finding a plane given three points.

For each of the two planes, we construct the map trace of the line and elevation points. If the lines intersect, we can use the actual lines. If the lines do not intersect, we can pick an arbitrary origin and draw lines parallel to the given lines and with the same plunge.

Next we construct elevation points on the lines. Lines through equal elevation points are structure contours on the desired plane. We can then find the strike and dip of the plane using methods we have already developed.

Examples

Find a Plane That Contains Two Intersecting Lines

 1. Given the two plunging lines, find the plane that contains them both. 2. Construct the map trace and elevation points for each line. 3. Draw structure contour lines through points of equal elevation. 4. Strike is simply the azimuth of the structure contours. Find dip by trigonometry or by a cross-section.

Find a Plane Parallel to Two Non-intersecting Lines

In the next example, the two lines don't intersect. For example, we might observe metamorphic foliation on two dipping joint planes, but not have enough surface relief to determine the dip. Also, we can't tell which foliation trace on one joint corresponds to any on the other joint. Nevertheless, we know the foliation has a strike and dip. We can measure the trend and plunge of the foliation trace on each joint and find the strike and dip of the plane parallel to both of them.

Note that the lines don't intersect and so the locations of the lines are immaterial. All that matters is the attitude of the lines and the plane parallel to both of them. Thus:

• We can pick an arbitrary point and draw lines parallel to the trends of the two measured lines.
• Actual elevations are not needed; we can simply measure depth below the intersection and need not interpolate.
 1. Given the two plunging lines, find the plane parallel to them both. 2. Pick a convenient point and draw lines parallel to the trends of the two lines. Construct elevation points for each line. Depth below the intersection will do. 3. Draw structure contour lines through points of equal elevation. 4. Strike is simply the azimuth of the structure contours. Find dip by trigonometry or by a cross-section.